Difference between revisions of "לקוטים מספר פראלוקא"

Introduction

Section One - Theoretical Arithmetic

• I say that the numbers 5 and 6 keep themselves in their square and their cube.

As: 5 times 5 is 25; so, the 5 is perceivable.

In its cube: 5 times 5 is 25, and 5 times 25 is 125; it is also perceivable.

Similarly you say for the 6: because, if you take the square of 6, it is 36; so, it is perceivable.

Also in its [cube]: because, 6 times 6 is 36, and 6 times 36 is 216; it is also perceivable in the cube.

The 1 also keeps itself in its square and its cube.

Therefore, Abraham Ibn Ezra said in parashat Shemot that the letters א [=1], ה [=5] , ו [=6] keep themselves in their square and their cube.

Perfect Numbers

3) Definition of the superabundant number: The discussion on the superabundant number, such as 12 or 24, for the sum of its parts exceeds over it, i.e. when you sum all the parts generated from 12, the result is 16, so its parts are more than its whole, and this number is called in their language "numero abbondante".

4) Definition of the deficient number: The discussion on the number, which the sum of its parts is less than the number [itself]; as eight, for the sum of its parts, which are the half, quarter and eighth, is only 7; and this number is called in their language "numero povero".

5) Definition of the perfect number: The discussion on the perfect number, which is the number, which the sum of its parts is the same as the given number no more and no less; as 6, for when you sum all its parts they are the same as it, since its half is 2, its third is 2, its sixth is 1 and the total is 6, as it is; and this number is called "numero perfetto".

6) The discussion about how we know if a given number is perfect or imperfect.

See: if in the rank of units it has 6 or 8, it might be a perfect number; if it does not have 6 or 8, it cannot be a perfect number.

The good thing is that you can find only one perfect number in each rank.

6 is a perfect number in the units.

28 in the tens.

All the numbers in the other ranks follow these.

The perfect number in the hundreds is 496 and you see that the 6 is in the units.

In the perfect number of the thousands you find the 8 in the units.

So is the rule for the rest of the ranks after them: one has 6 and the other 8.

7) The discussion about finding the way.

I say that God created his world in six days, because the number 6 is a perfect number and on the sixth day man was created, to indicate that he is the most perfect of all creatures.

I say that the way is as follows:

• We find the first: take one and add it to 2; it is 3.

See if the number 3 is composite: you see that it is prime.

So, multiply it by 2; it is 6 and this is the perfect number in the first rank.

• To find the second: take 1, 2, 4 and sum them; it is 7.

See if it is composite: you see that it is prime.

So, multiply 7 by the last number you took, which is 4; the result is 28 and this is the perfect number in the second rank.

• To find the third: take 1, 2, 4, 8 and sum them; it is 15.

You see it consists of 3 and 5.

So, take the other number, which is 1, 2, 4, 8, 16; the result is 31.

It is not composite.

Multiply it by the last number you took, which is 16; the result is 496 and this is the perfect number in the rank of hundreds.

• To find the fourth: take 1, 2, 4, 8, 16, 32 and sum them; it is 63.

It consists of 21 and 3.

So, take the other number, which is 1, 2, 4, 8, 16, 32, 64 and sum them; the result is 127.

Multiply it by the last number, which is 64; the result is 8 thousand and 128 and this is the fourth [perfect number].

• Example: we have the number 496. We wish to know if it is a perfect number and how many numbers it consists of, starting from one.

Divide 496 by 2; you receive 248.

Divide also 248 by 2; you receive 124.

Divide also 124 by 2; you receive 62.

Divide also 62 by 2; you receive 31.

Divide also 31 by 2; you cannot.

Therefore, divide the required number, which is 496 by 31; the result is 16.

Divide also 496 by 62, which is [double] 31; the result is 8.

Divide also 496 by 248, which is [double] 124; the result is 2.

Divide 496 by 496; the result is 1, because 496 is [double] 248.

Hence, 496 is a perfect number, because the division reaches one.

It consists of: 1, 2, 4, 8, 16, 32.

As you know the way mentioned below and this is the diagram:

All these numbers add up to 496.

• Another example of the number 8 thousand and 128: we wish to know if it is a perfect number and if it is a perfect number, how many numbers it consists of.

First, divide 8128 by 2; the result is 4 thousand and 64.

Divide also 4 thousand and 64 by 2; the result is 2 thousand and 32.

Divide also 2 thousand and 32 by 2; the result is one thousand and 16.

Divide also one thousand and 16 by 2; the result is 5 hundred and 8.

Divide also 5 hundred and 8 by 2; the result is 254.

Divide it also by 2; the result is 127.

Divide it also; you cannot.

Therefore, divide the number, which is 8 thousand and 128 by 127; the result is 64.

Divide also [8 thousand and 128] by 254, which is [double] 127; the result is 32.

Divide also 8 thousand and 128 by 5 hundred [and 8]; the result is 16.

Divide also 8 thousand and 128 by one thousand and 16; the result is 8.

• Example: we have ten proportional numbers of double ratio: the first is 1, 2, 4, 8, 16, 32, 64, 128, 256, 512.

I say that 1 is a root.

The 4, which is third, also has a root.

The fourth, which is 8, is a cube.

The fifth, which is 16, has a root; its root is 4, which is the third.

The sixth has no root.

The seventh, which is 64, has a cube root; its root is 4, which is the third.

The eighth has no root.

The ninth, which is 256, has a root, which is 16 and it is the fifth.

The tenth, which is 512, has a cube root; which is the fourth number, which is 8, because 8 is a cube root of 512.

• Example: 1, 3, 9, 27, 81, 243, 72[9].

I say that 1 that is the first has a root.

The second has no root.

The third, which is 9, has a root, because its root is 3.

The fourth, which is 27, has only a cube root.

The fifth, which is 81, has a square root.

The sixth has no root.

The seventh, which is 72[9], has a cube root; its root is 9, which is the third.

And so on.

See if there is 1 in the first rank of the cube, deduce that there is also 1 in the first rank of the root.

If you find 2, deduce that the root has 2 in the first [rank].

If 3, deduce that there is 7 in the first rank.

If 4, its root has also 4 in the first [rank].

If 5, its root has also 5 in the first [rank].

If 6, its root has also 6.

If 7, deduce that there is 3 in the first [rank].

If 8, deduce that the root has 2.

If 9, deduce that the root has 9 also.

Start from 2: double it; it is 4.

Double it; it is 8 and it is a cube.

Double it again; it is 16.

Double it again; it is 32.

Double it again; it is 64 and it is a cube.

Double it again; it is 128.

Double it again; it is 256.

Double it again; it is 512 and it is a cube.

Congruence of Squares

• To find the second congruent number we do as follows:

Take 2 and 3: as you did for the first, when you summed up 1 and 2, so for the second we add 2 to 3; it is 5.

Double it, as you did for the first; it is 10.

Then, multiply 2 by 3, as you did for the first; it is 6.

Multiply it by the reserved 10; it is 60.

Multiply it by 2, as you did for the first; it is 120 and this is the second congruent number.

To find its congruent:

Square 2; it is 4.

Square 3 also; it is 9.

Sum them; it is 13.

Square it; it is 169 and this is the congruent of the number 120, which is its congruent.

• To find the third [congruent number]:

Take 3 and 4; sum them up; it is 7.

Double it; it is 14.

Then, multiply 3 by 4; it is 12.

Multiply it by 14; it is 168.

Multiply it by 2; it is 3 hundred and 36 and this is the third congruent number.

29) The way to find a number, such that when adding ten to it, it is a square, and if you subtract 11 from it, it is a square.

Do as follows:

Add 10 to 11; it is 21.

Subtract one from it; twenty remains.

Divide it by 2; the result is ten.

Square it; it is 100.

Add to it the 11 that you wish to subtract; it is 111 and this is the required.

Because, if you subtract 11 from it, 100 remains; if you add ten to it, it is 121; and both are squares.

• If it is said: find a number, such that when you add 8 [to it], it is a square, and if you subtract 7 [from it], it is a square.

Do as you did:

Add 7 to 8; it is 15.

Subtract 1 from it; 14 remains.

Take its half; it is 7.

Square it; it is 49.

Add 7 to it; it is 56 and this is the required.

Because, if you add 8 to it, it is 64 and it is a square.

And, if you subtract 7, 49 remains and it is a square.

29) The way to find a number, such that when subtracting ten from it, a square remains, and when adding ten to it, it is a square.

30) The way to find two square numbers, such that one exceeds the other by 7.

31) The way to find a number, such that when adding it to five, it is a square and when subtracting it from five, it is a square.

32) The way to find a number, such that if you subtract 5 from it, it is a square and if you add 12 to it, it is a square.

Do as follows:

Add 5 to 12; it is 17.

Add one to it; it is 18.

Divide it by 2; it is 9.

Square it; it is 81.

Subtract 12 from it, as said; the result is 69 and this is the required.

Because, if you subtract 5 from 69, 64 remains and it is a square.

Also, if you add 12 to it, it is 81 and it is a square.

• Likewise, if it is said: if you add six to it, it is a square and if you subtract 10 from it, a square remains.

Add 6 to [10]; it is 16.

Add 1 to it; it is 17.

Divide it by 2; it is [8] and a half.

Square it; it is 72 and a quarter.

Subtract 6 from it; 66 and a quarter remains and this is the required.

Add 6 to it; it is 72 and a quarter and it is a square.

If you subtract 10 from it, 56 and a quarter remains and it is a square.

33) The way to find square numbers such that their sum is a square and if you add ten to them, it is also a square.

Do as follows:

Subtract 1 from 10; 9 remains.

Divide it by 2; 4 and a half remains.

Square it; it is twenty and a quarter and this is the square number

Now, take two numbers, such that when you take their squares and sum them up, the result is twenty and a quarter.

Say: 3 times 4 is 12; and 4 times 5 is 20.

Subtract 12 from twenty; 8 remains. Keep it.

Multiply also 4 by 4; it is 16; and 3 times 5; it is 15.

Sum them; it is 31.

Divide the reserved 8 by the root of the two numbers you assumed, whose sum is 25 and its root is 5.

Divide 8 by 5; the result is 1 and 3-fifths and this is one of the required numbers.

Divide also 31 by 5; the result is 6 and one-fifth and this is the second number.

Square 1 and 3-fifths; the result is 2 and 14 parts of 25.

Square 6 and one-fifth also; the result is 38 and 11 parts of 25.

Their sum is 41.

If you assume the other two numbers are: the first is 5 and the other is 12.

The [square] of 5 is 25 and the [square] of 12 is 144; their sum is 169 and its root is 13.

Multiply 5 by 4; it is 20; and 12 by 5; it is 60.

Subtract 20 from it; 40 remains.

Multiply 12 times 4; it is 48; and 5 by 5; it is 25.

Sum them; it is 73.

Divide 40 by 13; the result is 3 and one part of 13.

Divide also 73 by 13; the result is 5 and 8 parts of 13 and this is the required.

Because, the product of 3 and one part of 13 by itself is 9 and 79 parts of 169.

The product of 5 and 8 parts of 13 by itself is 31 and 90 parts of 169.

Their sum is 41 integers.

Do as follows:

We take the numbers 3 and 4, because their squares are 9 and 16; the sum is 25 and its root is 5.

Now, multiply [2] by 3; it is 6; and 4 by 4; it is 16.

Subtract 6 from 16; ten remains.

Then, multiply 4 by 2; it is 8; and 3 by 4; it is 12.

Their sum is twenty.

Then, divide 10 by 5; the result is 2 and this is the first.

To find the other, divide twenty by 5; the result is 4 and this is the second.

Now, multiply 3 by 4; it is 12.

Multiply also 4 by 2; it is 8.

Subtract it from 12; 4 remains. Keep it.

Then, multiply crosswise: 4 times 4 is 16 and 3 times 2 is 6.

Add them together; it is 22.

Then, take the partitore [divisor], i.e. 5, and divide the reserved 4 by 5; the result is 4-fifths and this is one of the numbers.

Divide also 22 by 5; the result is 4 and 2-fifths and this is the second number.

Multiply [4-fifths] by itself; it is 16 parts of 25.

Multiply also 4 and 2-fifths by itself; the result is 19 and 9 parts of 25.

Their sum is twenty.

Section Two - Practical Arithmetic

Check

Subtraction

Multiplication

Progressions

49) A way to know how much is the sum of the numbers beginning from one, by knowing half the sum.

• Example: we wish to know the sum of all the numbers from 1 to 12.

50) A way to sum from 1 by the succession of the numbers

It is the required, whether for even or for odd.

• Example: we wish to sum from 1 to 6.

• If you want to sum from 1 to 7.

51)

52)

53)

54)

57)

58)

59)

וכן אם תרצה לחבר כל המרובעים שהם מא' עד י"ב

60)

61)

62)

Motion Problems

69) Question: One man walks 32 miles every day; another man starts following him after 6 days and reaches him in 25 days.

I ask: how many miles did the latter walk each day?

אשאל כמה מילין הלך האחרון בכל יום

70) Question: a man walks 30 miles every day; at the end of 5 days another [man] starts following him and walks 35 every day.

I ask: in how many days will he reach him?

אשאל בכמה ימים ישיגנו

71) Question: a man walks, we do not know how many [miles] every day; at the end of 4 days [another] man starts following him and walks 35 every day. He reaches him at the end of twenty days.

I ask: how many [miles] does the former walk every day?

אשאל כמה הולך הראשון בכל יום

מתי יפגעו זה את זה

74) Question: A man has to walk ten miles. During the day he walks 2 [miles] and at night he goes back one [mile].

I ask: how many days will he walk?

אשאל בכמה ימים ילך

75) Question: A dog is sixty cubits away from a rabbit and 5 steps of the dog are equal to 7 of the rabbit.

I ask: when will the dog reach the rabbit?

‫^[37]אשאל מתי ישיג הכלב הארנבת

77) If you wish to sum up all the cubes from 1 to 12; i.e. 1, 8, 27, 64, 125.

78) If you wish to sum up all the squares from 1 to ten; i.e. 1, 4, 16, 25.

Question: to extract the root of fractions and integers or fractions alone.

• Example: we wish to extract the root of 9 parts of 16.

81) Question: find me 9 different squares such that their sum is a square number.

Cubic Roots

82) Question:

Fractions

83)

84) If you wish to multiply 30 by 5 sixths.

Multiply 30 by 5 then divide by 6 and the result is 25.

85) If you wish to subtract 3 fifths from 7 eighths.

• Common denominator: take a number that consists of 5 and 8, which is 40.

• Take its 3 fifths, they are 25.

• Take its 7 eighths, they are 35.

• Subtract 24 from 35, the remainder is 11 and they are 11 parts of the denominator, which is 40.

• As if you say: sum 2 thirds with 3 quarters

• Common denominator: take a number that consists of 3 and 4, which is 12 and it is the denominator.

• Take its 2 thirds, they are 8.

• And its 3 quarters, they are 9.

Sum 9 and 8, they are 17.

• Divide them by 12, which is the denominator and the result is 1 and 5 parts of 12.

86) If you wish to divide one half by one third.

• Common denominator: take the denominator, which is 6.

• Take its third, it is 2.

• And its half, it is 3.

• Divide 3 by 2, it is 1 and a half and it is the required.

• If you wish to divide 4 by 2 thirds.

87)

88) Which number will you sum with 23 so that the result will be 39?

• Subtract 23 from 39, the remainder is 16.

89) Which number will you sum with 6 and a half so that the result will be 13 and 2 fifths?

• Subtract 6 and a half from 13 and 2 fifths, the remainder is 6 and [9] parts of [10].

90) Which number will you divide by 5 so that the result will be 17?

• Multiply 5 by 17, the result is 85.

91) Which number will you divide by 4 and a seventh so that the result will be 2 and one eighth?

• Multiply 4 and a seventh by 2 and one eighth, the result is 8 and 45 parts of 56.

92) By which number will you divide 36 so that the result will be 9?

• Do as follows: divide 36 by 9, the result is 4 and this is the required.

93) By which number will you divide 12 and a quarter so that the result will be 3 and a half?

• Divide 12 and a quarter by 3 and a half, the result is 4 and a half.

94)

95) Question: how many parts of 18 there are in 5 ninths?

96)

97) Divide 7 and a half by a certain number so that the result will by 5 sixths of 17 and a half.

98)

99) By how many fifths should we multiply 3 and 2 thirds so that the result will be 7 and 3 quarters?

100) Divide 7 and a half by so and so ninths so that the result will be 2 and a [quarter].

101)

102)

103)

104)

105)

106)

107) Question:

108)

109) Question:

110) Question:

111) Question:

Ratios

112) Now we will discuss the ratios

The reason is that the double ratio is generated from the ratio of 1 to 2, for 3 is as 2 plus its half.

The triple ratio is generated from the ratio of 1 to 3, for it is greater than 2.

The quintuple is greater than the quadruple, because it is the ratio of 1 to 4 and the quintuple is from 1 to 5

113) Question: find me three proportional numbers, such that their product by each other is the same as their sum.

Do as follows: 1, 2, 3 are related, because the excess of the second over the first is 1 and the excess of the third over the second is 1.

Multiply 1 by 2; it is 2. Multiply it by 3; it is 6. Also the sum of 1, 2, 3 is 6.

114)

If the ratio is generated from the ratio of 1 to 2, it is called double ratio.

If it is generated from the ratio of 1 to 3, it is called triple ratio.

If it is generated from the ratio of 1 to 4, it is called quadruple ratio.

The sesquialter ratio is called the ratio of 3 to 2, for 3 is as 2 plus its half.

The sesquitertian ratio is [generated] from the ratio of 4 to 3, for 4 is as 3 plus its third.

The sesquiquartan ratio is [generated] from the ratio of 5 to 4, for 5 exceeds over 4 by its quarter.

The sesquiquintan ratio is [generated] from the ratio of 6 to 5, for 6 is as 5 plus its fifth added to it.

If the ratio is of 5 to 3 it is called superbipartient ratio, for 5 is as 3 plus 2.

If the number adds 3 over the other, it is called supertripartient, such as 8 to 5, for 8 is as 5 plus 3.

If it adds 4 [over the other], such as the ratio of 9 to 5, it is called superquadripartient.

If it added 5, such as the ratio of 11 to 6, it is called superquintipartient ratio.

115)

• Example: 1, 2, 3. The square of 1 is 1; the square of 2 is 4; the square of 3 is 9; together they are 14. If you sum 1, 2, 3, the result is 6. Divide 14 by 6; the result is 2 and one-third.

• Similarly, if you multiply 2, 4, 6, each by itself, and sum their squares, the result is 4 times and 2-thirds the sum of all three.

Example: multiply 2 [by itself]; it is 4. Multiply 4 [by itself]; it is 16. Multiply 6 [by itself]; it is 36. Sum them; the result is 56. Then, sum 2, 4, and 6; they are 12. It is twice the first.

• Likewise, if you multiply 4, 8, 12, each by itself, and sum their products, the result is 224. Divide 224 by their sum, i.e. 4, 8, and 12, which is 24; you receive 9 and one-third, which is twice the previous that was 4 and 2-thirds.

116) Question:

117)

118) If you have three proportional numbers and you sum them, then divide the sum by each of the three numbers, the quotients are also proportional numbers.

• Example: if we have 2, 4, and 8, they are proportional.

I say that if you sum them, they are 14.

If you divide 14 by 2, which is one of the quotients, the result is 7.

If you divide also 14 by 4, the result is 3 and a half.

If you divide 14 by 8, the result is 1 and 3-quarters.

These numbers are proportional.

119) If you sum three proportional numbers that are doubles or triples, always sum them, then divide the sum by each of the numbers [and the quotients] will always by the same.

• Example of the doubles: 2, 4, 8

Their sum is 14.

If you divide it by 2, the result is 7.

If by 4, [the result is] 3 and a half.

If by 8, [the result is] 1 and 3-quarters.

The same is done with other numbers that are doubles.

• Such as 6, 12, 24.

If you sum them, they are 42.

Divide it by 6; the result is 7 as at the beginning.

Divide it by 12; the result is 3 and a half.

Divide it by 24; the result is 1 and 3-quarters as at the beginning.

• Example for triples: 6, 18, 54.

Their sum is 78.

Divide it by 6; the result is 13.

Divide it by 18; the result is 4 and one-third.

Divide it by 24; the result is 1 and 4-ninths.

120)

Sum them; they are 18.

Then, divide 18 by 12; the result is 1 and a half.

Divide 18 also by 6; the result is 3.

You see that the ratio of 1 and a half to 3 is a double ratio.

If you multiply these two numbers, i.e. 1 and a half by 3, the result is 4 and a half.

Likewise, if you sum them, they are also 4 and a half.

Hence, their sum is the same as their product.

121)

122)

123)

124)

125)

126)

127)

128)

129)

130) If you have four proportional numbers, then the product of all of them by each other is the same as the [square of the product of the] first by the fourth.

• Example: 2, 4, 8, 16.

Now, multiply 2 by 4; it is 8; 8 by 8 is 64; and 16 times 64 is one thousand and 24.

If you multiply 2, which is the first, by 16, which is the fourth; it is 32 and its square is one thousand and 24.

131)

132)

133)

134)

135)

136)

137)

138)

139) Question:

140) Question:

141) If you have three proportional numbers, such that when you multiply the first by the second and the third, you also multiply the second by the two others, you multiply the third by the two others, you also multiply each by itself, and the result is 361.

How much are the three proportional numbers?

142) Produce from 13 three proportional numbers, such that when the first part is multiplied by the two others, the second is multiplied by the two others, and the third is multiplied by the two others, the total is 78.

143) Divide 19 into three proportional numbers, such that if you multiply the first by the second, then [the product] by the third, the result is 216.

144) Divide ten into three proportional numbers, such that the product of the first by the third is the same as 3 times the product of the first by the second.

Do as follows: since 3 is mentioned, take 1 for the first, 3 for the second, and for the third take it 3 times, which 9.

Sum them; it is 13.

Divide 10 by 13; the result is 10 parts of 13 and this is the first.

Multiply it 3 times; the result is 2 and 4 parts of 13.

Multiply it by 3; the result is 6 and 12 parts of 13 and this is the third.

145) Question: divide ten into four proportional parts, such that the product of the first part by the fourth is three times the product of the first part by the third.

Do as follows: we suppose that the first is 1, the second is 3, the third is 9 and the fourth is 27.

The total is 40.

Therefore, say: if 40 gives me 1, what will ten give you and how much will it add? The method is known.

We find that the first is a quarter, the second is 3-quarters, the third is 9-quarters, the fourth is 27-quarters.

146) Question: weigh for me with four stones up to 15.

Take 1, 2, 4 and 8; they are 15 and with these four you can weigh up to 15.

If a fifth is added to them one can weigh up to 31.

If you want another ratio, as if you say: weigh with four stones from 1 up to 40.

Take 1, 3, 9, 27 and they are 40.

If you add another stone, you can weigh up to 121 and so on.

If you want to weigh with four stones up to 35.

False Positions

147) Question: a man spent a third of his money and was left with 18.

How much was his money?

כמה ממונו

False Position:

Rule of Three:

148) Question: a man bought something but he did not know how much it is.

He knew that he earns 3⅓ soldi for every florin he spends.

The fund and the profit were 50 florins.

How much did he spend?

אמנם הוא יודע כי הוא מרויח ג' סולדי ושליש בעד כל פיורינו שפזר
ועולה הקרן והריוח עשרים פרחי‫'
כמה פזר

1 florin = 100 soldi

False Position:

florin

the profit of 30 florin: soldi = 1 florin

Rule of Three:

149) Question:

Rule of Three:

150) Question:

Rule of Three:

151) Question:

152) Question:

Rule of Three:

153) Question:

154) Question:

155) Question:

Euclidean Propositions

156) Euclid, Elements, Book II, proposition 1: If you have two numbers and you divide one of the two numbers, then you multiply each part by the second number, the total sum is the same as the product of the first number by the second.

• Example: the two numbers are 10 and 12 and we divide 12 to 2, 3, 5, 2, so their sum is 12.

As written in the first proposition of the second section of Euclid.

157) Euclid, Elements, Book II, propositions 2: If you have a number and you divide iy into parts as you wish, if you multiply each part by the divided number, then sum all the [products] they are equal to the divided number multiplied by itself.

• Example: we wish to divide 12 into three parts 5, 3, 4.

158) Euclid, Elements, Book II, proposition 3: If you divide any number into two parts, then multiply any of the two parts by the divided number and keep the product, so will be the result if you multiply the same part by itself, then add to it the product of the one part by the other.

• Example: we have 12 and you divide it to 8 and 4.

159) Euclid, Elements, Book II, proposition 4: If you divide any number into two and multiply each part by itself, you multiply also one [part] by the other and multiply [the product] by 2, then you sum all, the result is equal to the divided number multiplied by itself.

• Example: we divide 12 to 4 and 8.

160) Euclid, Elements, Book II, proposition 5: If you divide any number into two equal parts and into two unequal parts, the product of the equal parts one by the other is as the product of the unequal parts one by the other, when you add to it the [square] of the difference between the [equal] part and the [unequal part].

• Example: we divide 12 into two equal parts 6 and 6, and into two unequal parts 4 and 8.

161) Euclid, Elements, Book II, proposition 6: If you divide any number into two equal parts, then add an additional [number] to the divided number, multiply [the sum] by the additional [number] and add [the product] to the square of one of the parts, I say that it is equal to the square of half [the number] with the additional [number].

• Example: we have 12, you divide it to 6 and 6, then add to 6 whichever number you wish. Suppose that we wish to add 4 to 6.

162) Euclid, Elements, Book II, proposition 7: If you have a number and you divide it into two unequal parts, you multiply the greater part by itself and multiply also the divided number by itself, then sum both products, the result is equal to the product of the greater number by the divided number multiplied by two, then you add to it the square of smaller part.

• Example: we wish to divide 12 to 4 and 8.

163) Euclid, Elements, Book II, proposition 8: If you have a number and you divide it into unequal two parts, if you add one of the parts to the whole number, i.e. the divided number, [its square] is equal to the same part that you added multiplied by the whole number, then multiplied by four, when you add to it the square of the second part.

• Example: we wish to divide 12 to 4 and 8.

164) Euclid, Elements, Book II, proposition 9: If you divide any line into two equal parts and into two unequal parts, then you multiply each of the unequal parts [by itself] and sum them, they are twice [the sum of] the product of the equal parts plus the [square of] excess of the greater [part] over [half the number].

• Example: we wish to divide 12 into two equal parts 6 and 6 and into two unequal parts 8 and 4.

As explained in proposition 36 of the tenth [section] of Euclid.

165) Euclid, Elements, Book II, proposition 10: If you divide any number into two equal parts, then add another number to the divided number, square it and add to it the square of the [number] you added, it is twice [the sum of] the square of the additional [number] plus half [the number] with the square of half [the number].

• Example: we have 12, you divide it to 6 and 6, then add to 12 another number. Suppose that we add 8 to it.

166)

167)

168)

169) Euclid, Elements, Book II, proposition 4: If you have a number and you divide it into two unequal parts, I say that if you sum the [parts] that are multiplied by themselves, then you multiply the smaller [part] by the greater, multiply the product by 2 and sum it with the reserved, the result is equal to the [original] number multiplied by itself.

Divide a Number Problems

170) If you have a number and you divide it into two unequal numbers, multiply each one by itself and sum [the products], then multiply the smaller by the greater, multiply [the product] by 2 and subtract the product from the [sum of the] two products. The remainder is equal to the square of the greater number after you subtract the smaller from it.

Let the number be ten. Divide it into 8 and 2.

The sum of the products of these two numbers, i.e. 2 by 2, which is 4, and 8 by 8, which is 64, is 68. Keep it.

Then, multiply 2 by 8; it is 16.

Multiply it by 2; it is 32.

Subtract it from 68; the remainder is 36.

Likewise, [the square of] the excess of the greater [number] over the smaller, which is 6 by 6, is 36.

171) Divide a number into two unequal parts, divide the greater by the smaller, then multiply the quotient by the greater; the result is the same as the product of the greater by itself divided by the smaller.

The number ten. Divide it into 8 and 2.

Divide 8 by 2; the result is 4.

Multiply 4 by 8; it is 32. Keep it.

Then, multiply the greater by itself and divide the product of 8 by 8, which is 64, by the smaller number, which is 2; the result is 32 and it is equal to the reserved 32.

172)

173)

174)

Find a Number Problems

175) Question: find me two numbers, such that when you sum the squares of each by itself they are 58 and the product of the one by the other is 21.

Do as follows: multiply 21 by 2; it is 42.

Add it to 58; it is 100.

Now, take its root; it is 10.

Divide ten into two parts, such that the product of one by the other is 21.

One is 3, the other is 7; together they are 10.

Multiply [3] by itself; it is 9.

Multiply 7 by itself; it is 49.

Their sum is 58.

176) Question:

177) Question: find me two numbers, such that when they are multiplied by each other they are 16 and the excess of the greater multiplied by itself over the other multiplied by itself is 60. How much is the greater and which is the smaller?

17[8]) Question: find me [two] number[s], such that the one is 8 and 8 ninths more than the other and the product of one by the other is but one.

Do as follows: take a half of 8 and 8-ninths; it is 4 and 4-ninths.

Multiply it by itself; it is 19 and 61 parts of 81.

Add one to it, i.e. what you want the product of one by the other to be; the result is twenty and 61 parts of 81.

Take its root; it is 4 and 5-ninths.

Add it to 4 and 4 ninths; it is nine and this is the greater number.

The smaller is 4 and 5-ninths.

Subtract 4 and 4-ninths from it; the remainder is one-ninth.

You see that the greater exceeds the smaller by 8 and 8-ninths.

The product of one by the other is exactly one.

Divide a Number Problems

179) If you divide one number into two unequal parts:

As 10: divide it into 2 and 8. The sum of the squares of each by itself is 68. The difference between 2 and 8 is 6 and its square is 36. Subtract it from 68; the remainder is 32 and this is double the product of the greater number multiplied by the smaller [number]; because their product by each other is 16. Double it; it is 32.

הנה ההפרש שיש מב' לח' הוא ו' ומרובעו ל"ו
תוציאם מס"ח הנשאר ל"ב והוא כפל מכפל המספר הגדול כפול על הקטן
כי כפל זה על זה הוא י"ו כפלם ויהיו ל"ב

180) If you divide one number into two unequal parts:

Find a Number Problems

181) Find me two numbers, such that the sum of the two is 10 and the sum of their squares exceeds by 28 over their product by each other.

Do as follows: divide 10 into half; it is 5.

Multiply it by itself; it is 25.

Subtract it from 28; 3 remains.

Divide it by 3; the result is one and its root is one.

If you subtract 1 from 5, 4 remains and this is one of the two numbers.

If you add one to 5, it is 6 and this is the second number.

You see that the sum of the products of these numbers by themselves is 52.

Subtract from it the product of the greater by the smaller, which is 24; 28 remains as we said.

182) Question: divide ten into two parts, such that the square of the greater exceeds the square of the smaller by 60.

Do as follows: square ten; it is 100.

Subtract 60 from it; the remainder is 40.

Divide 40 by double 10, which is twenty; the result is 2 and this is the smaller part.

The complement to ten is 8 and this is the greater part.

183) Question: find me two numbers, such that when multiplied by themselves, then you sum these products and you a to them also the product of the smaller by the greater, the result is 84.

Square ten; it is 100.

Subtract 84 from it; the remainder is 16 and this is the product of the smaller by the greater.

Then, say: divide ten into two parts, such that their products by each other is 16.

You find that one part is 2 and the other is 8.

Divide a Number Problems

184)

185) Question: Divide ten into two unequal parts, [such that when] you divide the greater [part] by the smaller [part] and the smaller by the greater, then multiply [the sum] by twenty, the result is 85.

Do as follows: divide 10 into 2 parts, 2 and 8.

Divide 2 by 8; the result is a quarter.

Divide 8 by 2; the result is 4.

Together they are 4 and a quarter.

Then, divide 85 by twenty; the result is also 4 and a quarter.

Multiply 4 and a quarter by twenty; the result is 85.

186) Question:

187) Question:

Find a Number Problems

188) Question:

189) Question:

190) Question:

191) Question:

192) Question:

193) Question:

194) Question:

195) Question:

196) Question:

197) Question:

198) Question:

199) Question:

200) Question: produce from 29 two parts such that if you subtract the root of the smaller part from the root of the larger 3 will remain

201) Question: produce from 10 two parts such that if you subtract the root of the smaller part from the root of the larger 2 will remain

202) Question: produce from 29 two parts such that if you subtract the root of the smaller part from the root of the larger 3 will remain.

203) Question: produce from twenty two parts such that if you add to the smaller part twice its root, the square of the result is the same as the product of the two numbers multiplied by each other.

204) Question: produce from 13 two parts such that the sum of their roots is five.

205) Question:

206) Question:

207) Question:

208) Question:

209) Question:

210)

211) If you divide ten into two parts

• for example 6 and 4

גם חלק י' על ו' ויצא א' וב' שלישיות
ומחוברים הם ד' ושתות ושמרם

חלק ד' על ו' ויצא ב' שלישיות
ומחוברים הם ב' ושתות אחד

212) If you divide 12 by 2 and by 6

ושניהם מחוברים הם ק"ח
חלקם על כפל ג' על ו' שהם י"ח ויצא ו' והם שוים אל הו' השמורים

213) If you multiply a square by a square the result is a square

If you multiply a cube by a cube the result is a cube

214) Question:

215) Question: We have 3 proportional numbers, such that [the root of] the product of the first by the [third] is the mean.

How much is the first and the third?

כמה הוא הא' והג‫'

• Example: we have 2, 8, 32.

The product of 2 by 32 is 64; its root is 8 and this is the mean.